Symmetrically anharmonic oscillators

We propose a new nonrelativistic Pauli-type equation where some specific small relativistic terms are retained. With the confining potentials $V_{\infty }\mathrm{}\left(x\right)\mathrm{}$ approximated by the polynomials $V_m\mathrm{}\left(x\right)=g_0{x}^2+\cdots +g_m{x}^{2m+2}$, $g_m>0\mathrm{}$, the nonzero kinematical corrections $T_m-T_0$, where $T_m=h_0{p}^2+\cdots +h_m{p}^{2m+2}\simeq T_{\infty } ={({\mu }^2{c}^4+p^2{c}^2)}^{\frac{1}{2}}-\mu c^2$, are added to the anharmonic-oscillator Schrödinger equation, so that the $p-x$ symmetry typical for a harmonic-oscillator Hamiltonian is restored. As a consequence of this semirelativistic regularization, the analytic diagonalization of an entirely anharmonic Hamiltonian $H_{\mathrm{mm}}=T_m+V_m$ in terms of the $m\times m$-matrix continued fractions is obtained. Both the auxiliary fractions and the eigenstates converge very quickly. In the cases of the bounded spectrum of $H_{\mathrm{mm}}$ ($m=2q$), it is proved exactly for $q=1, 2, \mathrm{and} 3$ and conjectured for $q\ge 4$.